Hyperbolic Space

In Spring of 2016, I taught a new elective at IMSA called Modern Geometries. The course started with a deep dive into Euclid’s Elements followed by a careful analysis of the consequences when some of Euclid’s (often unstated) premises were challenged. We saw that by changing some of the assumptions about dimension, distance, and certain types of invariance, we could build geometries that were not only valid mathematical constructs but also useful models for describing parts of our world.

One of the geometries we looked at was hyperbolic geometry. To understand hyperbolic geometry, it helps to start with the flat Euclidean plane we are used to. If we add a bit of a uniform bend to the plane, it starts resembling a large sphere – think of Earth for example. The more bend we add, the smaller the sphere is. To form hyperbolic space, we simply bend the plane the other way.

What does it look like when we bend the other way? We could really use a physical model to sink our minds into. So, let’s think about planes and spheres in a buildable way. If we took a bunch of equilateral triangles and glued them together with $6$ meeting at a vertex, we get a flat plane tiling.

Planar Tiling

What if we only put $5$ triangles meeting at each vertex? Then we get a shape called an icosahedron, which is vaguely spherical.

Icosahedron

Since we want to “bend the plane the other way”, we’ll put more than $6$ triangles meeting at each vertex. The model below uses $7$.

What we see is a hyperbolic plane. To us, it looks overfull and scrunched up, but it is flat in the hyperbolic sense. Take notice of the triangle drawn in pencil on the surface. It is actually made up of straight (in the hyperbolic sense) lines. The problem is that because of the curving of the surface itself, straight lines tend to bulge inward a bit. (Similarly, if you draw a triangle on the surface of a sphere, its edges will bulge out a bit.)

This model is a great way to start thinking about the complexities of hyperbolic geometry. If you’re thinking of building your own model, I recommend that this project, like all construction projects, is best done with a lot of duct tape.

However, the triangular model was still a bit flimsy, and the rigidity of the individual triangles makes it hard to visualize what a smooth hyperbolic surface would look like. For that, I recommend the hyperbolic soccer ball. The regular soccer balls we’re used to kicking around have two hexagons and one pentagon meeting at each vertex, whereas a hyperbolic soccer ball has two hexagons and a heptagon. (Quick quiz: what happens if you glue three hexagons together at each vertex?)